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Baskakov operators

where , , , for all functions on for which the series converges. Here, is a sequence of functions defined on having the following properties for every , :

i) ;

ii) ;

iii) is completely monotone, i.e., ;

iv) there exists an integer such that , .

Baskakov studied convergence theorems of bounded continuous functions for the operators (a1). For saturation classes for continuous functions with compact support, see [a8]. For a result concerning bounded continuous functions, see [a3].

In his work on Baskakov operators, C.P. May [a6] took conditions slightly different from those mentioned above and showed that the local inverse and saturation theorems hold for functions with growth less than for some . Bernstein polynomials and Szász–Mirakian operators are the particular cases of Baskakov operators considered by May.

S.P. Singh [a7] studied simultaneous approximation, using another modification of the conditions in the original definition of Baskakov operators. However, it was shown that his result is not correct (cf., e.g., [a1], Remarks).

Motivated by the Durrmeyer integral modification of the Bernstein polynomials, M. Heilmann [a4] modified the Baskakov operators in a similar manner by replacing the discrete values in (a1) by an integral over the weighted function, namely,

where is a function on for which the right-hand side is defined. He studied global direct and inverse -approximation theorems for these operators.

Subsequently, a global direct result for simultaneous approximation in the -metric in terms of the second-order Ditzian–Totik modulus of smoothness was proved, see [a5]. For local direct results for simultaneous approximation of functions with polynomial growth, see [a5].